
The equality in the first definition holds as \(Y^{\lambda (\mathbb {G}_m)-s} = Y^{\lambda (\mathbb {G}_m)-ss}= Y_{\min } \setminus Z_{\min }\)  by the Hilbert–Mumford criterion for \(\lambda (\mathbb {G}_m)\)  and the assumption that the linearisation is adapted. This is an open set in \(Y\) , as the assumption REF  implies that \(U Z_{\min } \subset Y_{\min }^0\)  is a closed subscheme (cf. [1]}).

In [2]}, the \(\widehat{U}\) -stable set is denoted \(Y^{\widehat{U}-s}_{\min +}\)  rather than \(Y^{\widehat{U}-s}\) , but we have simplified the notation as we will not introduce any other stable sets (see [2]}, [1]} for a discussion of various other notions of stability defined using invariant sections).

Our definition of the \(H\) -stable set differs slightly from that given in [1]}. However, under the assumption REF , both these notions coincide (see Theorem REF  below).

These definitions are functorial with respect to equivariant closed immersions \(Z \hookrightarrow Y\) ; that is, \(Z^{H-s} = Z \times _Y Y^{H-s}\) .


